COURSE OBJECTIVES LIST: GEOMETRY Geometry Honors is offered. PREREQUISITES: All skills from Algebra I are assumed. A prerequisites test is given during the first week of class to assess knowledge of these prerequisite skills and to locate deficiencies. COURSE BOOK DESCRIPTION: There are several ideas that intertwine in geometry: the study of properties of geometric figures (such as lines, triangles, quadrilaterals and circles); the maturation of inductive and deductive logical skills; the development of area and volume concepts; the exploration of the beautiful interaction between geometry and algebra. The Geometer s Sketchpad is used to give girls extensive hands-on experience with the mathematical concepts and to encourage experimentation. An honors section more deeply explores the rigors of mathematical proof. The course objectives are elaborated as follows. The order in which the objectives are listed is not necessarily the order in which they will be taught. Note that different texts may use different notation. PROPERTIES OF GEOMETRIC FIGURES: GEOM1. GEOM2. GEOM3. GEOM4. GEOM5. intuitive understanding of points, lines, and planes; collinear and coplanar; concept of intersection notation for points, lines, and planes: points (capital roman, A); lines (lowercase script l, or AB); planes (uppercase script P, or three noncollinear points ABC) space (the collection of all points); geometric figure (a subset of space) dimension concept: one, two, and three-dimensional; concept of perimeter for two-dimensional objects distinguish between CONGRUENCY (same size and shape) and SIMILAR- ITY (same shape, but not necessarily the same size); use proportions or scaling factors to find measurements in similar figures LINE SEGMENTS, LINES and RAYS: LINE1. LINE2. LINE3. LINE4. LINE5. distinguish between a line segment (AB) and the length of the line segment (AB). A line segment is a geometric figure; its length is a number. perpendicular bisector of a segment: definition (cuts the segment into two equal pieces and is perpendicular to the segment); important property as the set of all points equidistant from the endpoints concept of distance between a point and a line; distance between parallel lines definition of a ray; notation for a ray ( AB) terminology for lines: parallel, perpendicular, skew, transversal Miss Hall s School Course Objectives List: Geometry page 1 REV: 08/02
LINE6. lines cut by a transversal: interior and exterior angles; alternate interior angles; corresponding angles; consecutive interior angles LINE7. (test for parallel lines) The following are equivalent: two lines are parallel a pair of alternate interior angles are congruent a pair of alternate exterior angles are congruent a pair of consecutive interior angles are supplementary a pair of corresponding angles are congruent ANGLES: ANGLE1. ANGLE2. ANGLE3. definition of angle; notation for angles ( ABC) terminology for angles: vertex; interior; exterior; acute, obtuse, and right angles; degree measure; adjacent angles; vertical angles (and congruency theorem); linear pair; complementary and supplementary; angle bisector angle bisector: definition (cuts the angle into two angles of equal measure); important property as the set of all points equidistant from the sides of the angle TRIANGLES: TRI1. terminology for triangles: vertex (vertices); sides; right triangle (legs, hypotenuse); isosceles; equilateral; scalene; median; altitude TRI2. the base angles of an isosceles triangle are congruent TRI3. The sum of the (measures of the) angles in any triangle is 180. TRI4. TRI5. TRI6. TRI7. TRI8. Exterior Angle Theorem for a triangle congruent triangles: CPCTC (Corresponding Parts of Congruent Triangles are Congruent); SSS, SAS, ASA, AAS congruency theorems; there is not a SSA congruence theorem be able to draw a counterexample inequalities for sides and angles of a triangle: Let s 1 and s 2 be the lengths of two sides in a triangle, with opposite angles θ 1 and θ 2 respectively. Then, s 2 > s 1 iff m(θ 2 ) > m(θ 1 ). Triangle Inequality Theorem: the sum of the lengths of any two sides in a triangle is greater than the length of the third side. In other words; the shortest distance between two points is a straight line. additional exploration of triangles, to provide interest, practice, and depth: e.g., similarity theorems for triangles; or, the Hinge Theorem (inequalities involving two triangles; given two triangles with two congruent sides, the side opposite the larger included angle is longer than the side opposite the smaller included angle) Miss Hall s School Course Objectives List: Geometry page 2 REV: 08/02
TRI9. TRI10. TRI11. TRI12. the Pythagorean Theorem and its proof (this is also covered in Algebra I; try to give a different proof than they saw in the previous course) lengths of sides in two special triangles: the 30-60 -90 triangle and the 45-45 -90 triangle. right triangle definitions of sine, cosine, and tangent, to approximate lengths of sides in right triangles; calculator usage, with awareness of angle mode calculator usage of cos 1, sin 1, and tan 1 to approximate unknown angles in triangles QUADRILATERALS: QUAD1. QUAD2. QUAD3. QUAD4. quadrilaterals (4-sided polygons): parallelogram; rectangle; rhombus; square; trapezoid concept of Venn diagram: the diagram summarizing the relationship between quadrilaterals, parallelograms, rhombi, rectangles, and squares may be useful (test for parallelograms) The following are equivalent for a quadrilateral. (That is, given a quadrilateral, if one of the following statements is true, then they are all true; if one is false, then they are all false.) it is a parallelogram both pairs of opposite sides are congruent both pairs of opposite angles are congruent the diagonals bisect each other one pair of opposite sides is both parallel and congruent additional exploration of quadrilaterals, to provide interest, practice, and depth: e.g., the median property of trapezoids: its length is the average of the length of the bases CIRCLES: CIRCLE1. CIRCLE2. CIRCLE3. CIRCLE4. CIRCLE5. CIRCLE6. CIRCLE7. terminology for circles: center; radius (radii); diameter; chord; arc; central angle; semicircle; tangent to a circle; secant line; sector making pie charts (circle graphs) the irrational number π as the ratio of the circumference to the diameter in any circle; common approximations to π as a fraction ( 22 7 ) and decimal (3.14) formula for the circumference of a circle relationship between inscribed angles and intercepted arcs A line is tangent to a circle at point P if and only if it is perpendicular to the radius at P. Some additional theorems concerning tangent and/or secant lines should be investigated, to give depth to the coverage of circles. However, the instructor may pick and choose as desired. algebraic description of a circle; writing equations of circles; graphing circles that are already in the form (x h) 2 + (y k) 2 = r 2 Miss Hall s School Course Objectives List: Geometry page 3 REV: 08/02
POLYGONS: POLY1. POLY2. terminology for polygons: triangle, quadrilateral, pentagon, hexagon; regular polygons; convex figures additional exploration of polygons, to provide interest, practice, and depth: e.g., perimeter/area formulas for regular polygons; or, formula for the sum of interior and exterior angles in convex polygons LOGICAL SKILLS: The study of logic and mathematical language issues that was begun in Algebra I is firmed up and extended. Girls are responsible for the logic and math language concepts from Algebra I (those developed in One Mathematical Cat, Please! ). The Cat book is available on the web: http://fishcaro.crosswinds.net/cat book.htm Girls should apply their logical skills to a variety of proofs involving geometric figures. An Honors section will do more proofs, and proofs of higher complexity. The Sample Questions illustrate the difference in level of proof required for Honors and non-honors sections. LOGIC1. LOGIC2. LOGIC3. LOGIC4. LOGIC5. LOGIC6. inductive reasoning: conjecture (educated guess); making conjectures; counterexample (a specific example that shows that a conjecture is not true) deductive reasoning: the two-column proof is easier for beginners; Honors sections may also work with the paragraph-style proof truth tables (i.e., the definitions) and verbalizations for the basic sentence connectives: A and B, A or B, A B, A B, not A. equivalent forms of conditional sentences (implications): If A, then B (or: B, if A) A implies B A B Whenever A, B (or: B, whenever A) hypothesis; conclusion; vacuously true; converse; contrapositive (equivalences for implications and their verbalizations) The following are equivalent: A B not B not A (not A) or B Direct proof of an implication; proof of an implication by contrapositive. NEGATING SENTENCES (in a nice way): sentence a = b negation a = b a > b a b a b a > b, etc. A B A and (not B) A and B (not A) or (not B) A or B (not A) and (not B) In particular: note that the negation of an implication is not an implication. Miss Hall s School Course Objectives List: Geometry page 4 REV: 08/02
AREA AND VOLUME CONCEPTS: AREA1. AREA2. AREA3. AREA4. AREA5. AREA6. basic area concept: What is a square unit? Be able to show specified amounts of area, e.g., one square inch, 3 cm 2, one square blah (where blah is some unit of length) formulas for areas of: rectangle, parallelogram, triangle, circle, trapezoid terminology: simple closed curve, non-simple closed curve, cylinder, prism, polyhedron, sphere, cone, pyramid formula: surface area of a right circular cylinder formulas: volume of a cylinder; volume of a cone formulas: volume and surface area of a sphere ADDITIONAL EXPLORATION: To provide interest, practice, and depth, time should be spent exploring a topic of interest to teacher/students. For example: INTERACTION BETWEEN GEOMETRY AND ALGEBRA: Choose one conic section (ellipse, hyperbola, or parabola) and explore the relationship between its geometric (defining) properties, and its algebraic representation once a coordinate system is introduced. TRANSFORMATIONS: Explore translations, rotations, reflections, and dilations. NON-EUCLIDEAN GEOMETRIES GEOMETER S SKETCHPAD: GSK1. Girls should be comfortable working with the Geometer s Sketchpad. It should be used regularly throughout the course. CONSTRUCTIONS and SKETCHING SKILLS: CONST1. Draw a box, showing hidden lines and giving the illusion of three dimensions. Only a compass and straightedge are used for these constructions. Girls must be able to explain why each construction works. CONST2. CONST3. CONST4. CONST5. CONST6. CONST7. copying a segment bisecting a segment copying an angle bisecting an angle constructing parallel lines dropping a perpendicular from a point to a line Other constructions are optional. Miss Hall s School Course Objectives List: Geometry page 5 REV: 08/02